Environmental Engineering Reference
In-Depth Information
where C
∗
2
is a coefficient which depends on the DM halo to stellar mass ratio
R
, fractional
scaling radius Β, and radial orbital anisotropy.
Following the same procedure adopted in an earlier attempt (Ciotti et al., 1996), by solv-
ing Jeans equation for a two-component system, the aperture projected velocity dispersion
for the stellar component is obtained as:
2
Π
M
∗
p
(
R
ap
)
R
AP
Σ
ap
(
R
ap
) =
Υ
∗
I
(
R
)
Σ
p
(
R
)
R
d
R
(27)
0
where M
∗
p
(
R
ap
)
is the projected stellar mass inside the aperture radius R
ap
, I
(
R
)
is the
projected luminosity profile, Σ
p
(
R
)
is the projected velocity dispersion profile and
Υ
∗
is the
stellar mass-to-light ratio, assumed constant throughout the galaxy. For MVPE configura-
tions, with the position, Σ
ap
=
Σ
o
, the coefficient C
∗
2
, which comprises the effects of the DM
halo (leaving aside non-homology and radial orbital anisotropy from our analysis), can be
expressed as a function of either
Β, by substituting Eq.(20) and Eq.(21) into Eq.(26).
Accordingly, the following relations hold:
R
or
C
∗(HH)
F
(HH)
Β
) =
F
(HH)
R
(
=
(
R)
(28)
Β
2
C
∗(HP )
F
(HP )
Β
) =
F
(HP )
=
(
R
(
R)
(29)
2
Β
where the functions are calculated for spherical, isotropic models. The result is plotted in
Fig.5: being
Β, C
∗
2
also does with both
Β. This
is going on the right way, because larger C
∗
2
means, as deduced from the FP edge on view,
less massive galaxies, and this is compatible with wide amounts of DM (i.e., larger
R=
M
h
/M
∗
), and relatively smaller structures (i.e., larger Β
=
R
h
/R
∗
, which means smaller
R
∗
); the vice versa holds true also.
The
R
monotonically increasing with
R
and
Β relation plotted in Fig.4 has been splitted in Fig.5 by replacing C
∗
2
with either
R
-
R
(right panel), respectively. Values of C
∗
2
within the shaded region, corre-
spond to MVPE configurations with unreasonably high
Β (left panel) or
R
>25 for HH models).
Other models with halo inner logarithmic slopes ranging between 0 (P profiles) and 1 (H
profiles), would correspond to MVPE configurations with more reasonable
R
values (
R
values.
4.1.
MVPE Configurations and the FP
The most diffuse representation of the FP for ETGs writes:
log
R
e
=
A
log
Σ
0
+
B
I
e
+
C
(30)
where A, B and C, are constant coefficients (for each wavelength filter bandpass) derived
by means of a multiple regression fit of the effective radius R
e
(the radius of the circle
encircling half the total galaxy luminosity), the central projected velocity dispersion, Σ
o
,
and the average effective surface brightness within the effective area in flux units,
I
e
.
The deviation of the A coefficient from the value of 2 predicted by the Virial Theorem
is commonly known as the “tilt problem” of the FP. Several attempts have been made to
understand the origin of the observed tilt. Among them we recall here: 1) the radial orbital
anisotropy (see, e.g., Nipoti et al., 2002) in the velocity distribution of ETGs; 2) the weak
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